Transcript Rays and Angles

Aim: What are rays and angles?
Do Now:
1) Given points A(8, -5) and B(0, -11):
a) Determine the coordinates of the midpoint
of AB . Ans. M(4, -8)
b) Determine AB. Ans. 10
2) Sketch angles with the following measures:
a) Less than 90
b) Exactly 90
c) Greater than 90
d) Exactly 180
Geometry Lesson: Rays, Angles
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Def: Rays A ray is a part of a line that consists of an
endpoint, and all points on one side of the
endpoint.
A
P
PA = “Ray PA”
Def: Opposite Rays
Opposite Rays are two rays of the same line with a
common endpoint and no other points in common.
B
P
A
PA and PB are opposite rays
Geometry Lesson: Rays, Angles
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Def:Angle: An angle is the union of two rays having
the same endpoint.
side
A
x
B
vertex
C
side
Naming Angles:
a) Three capital letters, with vertex in the middle:
ABC or CBA
b) Single lowercase letter or number inside the angle: x
c) Use the name of the vertex angle if it’s
B
the only angle at that vertex:
Geometry Lesson: Rays, Angles
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Angle Measure: The measure of an angle is the
number of degrees in the angle.
mABC
A
B
25°

25°
mABC  25°
C
Def: Congruent Angles are angles having equal measure.
If ABC  EFG , then mABC  mEFG
Q: Which of the following angles are congruent?
A)
A
B)
45°
C)
45°
C
45°
Ans.
B
Geometry Lesson: Rays, Angles
A  B  
C
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Def: Straight Angle A straight angle is the union of
A
O
two opposite rays.
Straight angles have a
B measure of 180°
mAOB  180°
Def: Right Angle: A right angle has a measure of 90°
P
M
mPMQ  90°
Q
Geometry Lesson: Rays, Angles
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Def: Acute Angle: An acute angle has a measure greater
than 0° and less than 90 °.
X
Y
0°  mXYZ  90°
Z
Def: Obtuse Angle: An obtuse angle has a measure greater
than 90° and less than 180°.
Q
X
R
90°  mQXR  180°
Geometry Lesson: Rays, Angles
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Def:
Perpendicular Lines are two lines that intersect to form
right angles.
m
ml
Are lines m and l perpendicular?
l
NOT UNLESS SPECIFIED
BY THE GIVEN INFO OR A
BOX IN THE DIAGRAM !!!!
L
HK  JL
Right Angles: HJL KJL
Straight Angle:
H
J
HJK
K
Geometry Lesson: Rays, Angles
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Addition/Subtraction
of Angles:
If several angles share a common
T
P
20°
S
vertex, we can write addition and
subtraction expressions using the
names of the angles.
38°
R
TSP  PSR ?TSR
TSR  PSR 
? TSP
mTSP  mPSR  mTSR mTSR  mPSR  mTSP
20° + 38° = 58°
58° - 38° = 20°
Geometry Lesson: Rays, Angles
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Def: Angle Bisector: The bisector of an angle is a ray
that divides the angle into two
congruent angles.
EE
40°
80°
FF 40°
P
How do we bisect EFG ?
Ans.: Make a ray from the
vertex that divides it in half.
G
G
Conclusions:
If FP bisects EFG , then:
A) EFP  PFG Or mEFP  mPFG
1
B) mEFP  mEFG
2
1
C) mPFG  mEFG
2
Geometry Lesson: Rays, Angles
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M
Ex
#1Bisector Examples
Angle
PT bisects NPR
a) State a pair of
congruent angles.
NPT  TPR
N
P
T
R
S
b) Is MPT  TPS ?
There is no way to tell from the given info.
c) If mNPT  9 x  5 and mTPR  6 x  7 , find x.
x=4
Geometry Lesson: Rays, Angles
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Ex# 2 Simple 2-Column Proof:
Given: DB bisects ADC
Prove: mADB  mBDC
D
A
B
C
Statement
Reason
1) DB bisects ADC 1) Given
2) ADB  BDC
2) Def. of Angle Bisector
3) mADB  mBDC 3) Def. of Congruent Angles
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